One club I belong to has a rule that you must fire from a bench and the forend of the rifle must be on a rest.
The club supplys sand bags and has cushioned wooden rests available. I have been shooting off the bags at the club.
Bipods are OK to use and I have been thinking about getting one for club use.
Can I expect much change in elevation with a bipod on a non freefloat rifle.

Depends.  There should/could be some shift.  It really depends on how hard you are or are not resting your cheek on the back of the stock.  All kinds of weight impulses can change the rifle's trajectory when you have a bipod attached with a non free float.  I know that it would be frustrating to me as a shooter if my groups were shifting and I didn't know exactly why... I think that throwing a free float on there will reduce your chances of having that kind of frustration at the range if you are shooting for groups.

I find bags to be a more solid rest than a bipod..

i find my atlas is pretty twitchy regardless how tight i have it setup, it helps a lot to have a rear bag

It's all I use

For precision shooting there's a lot to be said for a soild rest (front/rear) as well as a solid bench to get the best out of your rifle. A bi pod isn't anywhere close.

A club that mandates a rest?

Yes they mandate a rest because some A holes were fireing from the hip and shots were going over the berm and the club got complaints.
Old club and when the range was set up there was nothing but woods beyond the berm. Now there are comercial buildings back there.

I must be the mis-fit here.  I have better accuracy when using a bipod.  The reason being is because I load the bipod which makes the rifle recoil straight back.

^^^ This, you need to lean into the bipod and "preload" it otherwise the rifle will simply bounce around.
 

When I use a bipod, I make sure to have the legs on something soft (but not "bouncy"), be it a coat or tee shirt . . . I've even used carpet padding! I've found though, that (as the poster above me said, I think. "Preloading" may refer to the pressure you put on the legs, too") if I brace the legs with sand bags, much of the bounce is eliminated. Or, at least, my groups tighten up, so I think that's what it is.

Sandbags are super stable & supportive, but what I do now is support my rifle/carbine on my bug-out-bag. I found this . . . "thing" with MOLLE straps and two parallel foam pads sewn in. The hand-guards snuggle right in there, between the pads and it makes for an excellent shooting platform. I figure it's good for consistency, too - should I even need to go prone. You know?

As for the sandbag behind the buttstock . . . eh. That's not my thing. I might go for an adjustable monopod though, although, that too has drawbacks.

The best advice anyone can give you? Get a FF tube/quadrail (unless you're shooting a retro, or something which "prohibits" it). The difference in performance is apparent with your first grouping.